.
The website for this book provides some artificial data suitable for fitting a principal curve. There are 50 observations of a bivariate variable, and each coordinate has been standardized. Denote these data as
x1, . . . ,
x50.
a.
Plot the data. Let ˆf(0) correspond to the segment of the line through the origin with slope 1 onto which the data project. Superimpose this line on the graph. Imitating the top left, show how the data project onto ˆf(0).
b.
Compute
τ(0)(x
i) for each data point
x
i. Transform to unit speed. Hint: Show why the transformation
aTx
i
works, where
a
c.
For each coordinate of the data in turn, plot the data values for that coordinate (i.e., the
xik
values for
i
= 1, . . . ,
50 and
k
= 1 or
k
= 2) against the projection index values,
τ(0)(x
i). Smooth the points in each plot, and superimpose the smooth on each graph. This mimics the center and right top panels .
d.
Superimpose ˆf(1) over a scatterplot of the data, as in the bottom left panel.
e.
Advanced readers may consider automating and extending these steps to produce an iterative algorithm whose iterates converges to the estimated principal curve. Some related software for fitting principal curves is available as packages for R (www.r-project.org).